Analysis of Higher-Order Ising Hamiltonians

TL;DR

This paper addresses the challenge of scaling higher-order Ising models by introducing IsingSim, a framework that decouples Ising spins from gradients and employs bidirectional differentiation for hyperedge functions. It formalizes a higher-order Ising Hamiltonian $\mathcal{H}(x)=\sum_{e\in E} w_e f_e(\{x_l: l\in e\})$, uses Walsh-Fourier expansion to represent hyperedge constraints, and analyzes ground-state relaxations that connect discrete satisfiability with continuous minima. The authors develop a scalable evaluation via convolution-based reductions and present three gradient computation strategies—exact, estimated, and Moreau-envelope-based—implemented with cumulative convolution and sampling. Experiments on parity-learning-with-error problems and 2-XOR benchmarks show that Type I spins often converge better, and that $\nabla \mathcal{H}$ and $\tilde{\nabla} \mathcal{H}$ are practical for hardware-oriented designs, while Moreau-based gradients can locate global minima at higher cost. Overall, IsingSim provides a flexible, quantitative tool for guiding higher-order Ising-machine design and evaluation.

Abstract

It is challenging to scale Ising machines for industrial-level problems due to algorithm or hardware limitations. Although higher-order Ising models provide a more compact encoding, they are, however, hard to physically implement. This work proposes a theoretical framework of a higher-order Ising simulator, IsingSim. The Ising spins and gradients in IsingSim are decoupled and self-customizable. We significantly accelerate the simulation speed via a bidirectional approach for differentiating the hyperedge functions. Our proof-of-concept implementation verifies the theoretical framework by simulating the Ising spins with exact and approximate gradients. Experiment results show that our novel framework can be a useful tool for providing design guidelines for higher-order Ising machines.

Figures (3)

• Figure 1: Left. Encoding a combinatorial problem, e.g., parity learning with error problem, into a higher-order Ising model. Right. The overall flow of IsingSim.
• Figure 2: The gradient descent trajectories of two coupled a. type I, b. type II ($p=1$), and c. type III Ising spins under XOR(x1,x2). Black crosses are the initial guesses. The crosses with other colors are the convergence points.
• Figure 3: Success rates at each gradient step using a. $\nabla\bar{H}$, b. $\tilde{\nabla}\bar{H}$, and c. $\tilde{\nabla}u$. * denotes the success rates on PLE problems with 8 parity bits.

Theorems & Definitions (17)

• Example 1
• Definition 1: Higher-Order Ising Model and Hybrid SAT
• Lemma 1: Reduction
• Theorem 1: Walsh-Fourier Expansion o2021analysis
• Example 2
• Theorem 2: Type I Relaxation kyrillidis2020fouriersatcen2023massively
• proof
• Corollary 1: Type II Relaxation
• proof
• Corollary 2: Type III Relaxation